Class 11 Maths Chapter 1 Sets Exercise 1.3 Solutions in English Medium
EXERCISE 1.3
EXERCISE
1.3
Class 11 Maths Chapter 1 Exercise
1.3 Question 1
1. Make
correct statements by filling in the symbol ⊂ or ⊄ in the blank spaces:
(i)
{ 2, 3, 4 } . . . { 1, 2, 3, 4, 5 }
(ii)
{ a, b, c, } . . . { b, c, d }
(iii)
{ x : x is a student of Class XI of your
school } . . . { x : x student of our school }
(iv)
{ x : x is a circle in the plane } . . . { x
: x is a circle in the same plane with radius 1 unit }
(v)
{ x : x is a triangle in a plane } . . . {
x : x is a rectangle in the plane }
(vi)
{ x : x is an equilateral triangle in a
plane } . . . . . . { x : x is a triangle
in the same plane }
(vii)
{ x : x is an even natural number } . . . .
{ x : x is an integer }
Solution:-
(i)
{
2, 3, 4 } ⊂ { 1, 2, 3, 4, 5 }
(ii)
{ a, b, c, } ⊄ {
b, c, d }
(iii)
{ x : x is a student of Class XI of your
school } ⊂ { x : x student of our school }
(iv)
{ x : x is a circle in the plane } ⊄ {
x : x is a circle in the same plane with radius 1 unit }
(v)
{ x : x is a triangle in a plane } ⊄ { x : x is a rectangle in the plane }
(vi)
{ x : x is an equilateral triangle in a
plane } ⊂ { x : x is a triangle
in the same plane }
(vii)
{ x : x is an even natural number } ⊂ {
x : x is an integer }
Class 11 Maths Chapter 1 Exercise
1.3 Question 2
2. Examine
whether the following statements are true or false :
(i)
{ a, b } ⊄ { b, c, a }
(ii)
{ a, e } ⊂ { x : x is a vowel in the English alphabet }
(iii)
{ 1, 2, 3 } ⊂ { 1, 3, 5 }
(iv)
{ a } ⊂ { a, b, c }
(v)
{ a } ∈ {
a, b, c }
(vi)
{ x : x is an even natural number less
than 6 } ⊂ { x : x is a natural number which divides 36 }
Solution:-
(i)
False, each element of { a, b } is
also an element of { b, c, a } .
(ii)
True , a, e are two vowels of the
English alphabet .
(iii)
False, 2 ∈ { 1, 2, 3 }; however, 2 ∉ {
1, 3, 5 }
(iv)
True, Each element of { a } is also an
element { a, b, c }
(v)
False, the element of { a, b, c } are a,
b, c . Therefore { a} ⊂ { a, b, c }
(vi)
True, { x : x is an even natural number less than 6 }
= { 2, 4 } ⊂ { x : x is a natural number which divides 36 } = { 1, 2, 3, 4, 6,
9, 12, 18, 36 }
Class 11 Maths Chapter 1 Exercise
1.3 Question 3
3. Let
A = { 1, 2, { 3, 4 }, 5 }. Which of the following statements are incorrect and
why ?
(i)
{ 3, 4 } ⊂
A
(ii)
{ 3, 4 } ∈ A
(iii)
{ { 3, 4 } } ⊂
A
(iv)
1 ∈ A
(v)
1 ⊂ A
(vi)
{ 1, 2, 5 } ⊂
A
(vii)
{ 1, 2, 5 } ∈ A
(viii)
{ 1, 2, 3 } ⊂ A
(ix)
Φ ∈ A
(x)
Φ ⊂ A
(xi)
{ Φ } ⊂ A
Solution:-
Given
A = { 1, 2, { 3, 4 }, 5 }
(i)
The statements { 3, 4 } ⊂ A is incorrect because 3 ∈ { 3, 4 }; however 3 ∉ A .
(ii)
The statements { 3, 4 } ⊂ A is incorrect because { 3, 4 } is element of A
.
(iii)
The statements { { 3, 4 } } ⊂ A is incorrect because { 3, 4 } ∈ { { 3, 4 } } and { 3, 4 } ∈ A.
(iv)
The statement 1 ∈ A is correct because 1 is
an element of A.
(v)
The statement 1 ⊂ A is incorrect because an element of a set can
never be a subset of itself .
(vi)
The statement { 1, 2, 6 } ⊂ A is
correct because each element of { 1, 2, 5 } is also an element of A.
(vii)
The statement { 1, 2, 5 } ∈ incorrect
because { 1, 2, 5 } is not an element of A.
(viii)
The statement { 1, 2, 3 } ⊂ A is
incorrect because 3 ∈ { 1, 2, 3 }; however 3 ∉ A .
(ix)
The statement Φ ∈ A is incorrect because Φ
is not an element of A .
(x)
The statement Φ ⊂ A is correct because Φ is a subset every set .
Class 11 Maths Chapter 1 Exercise
1.3 Question 4
4. Write
down all the subsets of the following sets
(i)
{ a }
(ii)
{ a, b }
(iii)
{ 1, 2, 3 }
(iv)
Φ
Solution:-
(i)
The subsets of { a } are Φ and { a } .
(ii)
The subsets of { a, b } are Φ , {a}, {b}
and { a, b } .
(iii)
The subsets of { 1, 2, 3 } are Φ, {1}, {2},
{3}, {1, 2}, {2,3}, {1, 3} and { 1, 2, 3
} .
(iv)
The only subsets of Φ is Φ .
Class 11 Maths Chapter 1 Exercise 1.3 Question 5
5. How
many elements has P(A), if A = Φ ?
Solution:-
We know that if A is a set
with m elements i.e, n(A) = m, then n [P(A)] = 2m
If A = Φ ,
then n(A) = 0.
∴ Hence, P(A) has one element .
Class 11 Maths Chapter 1 Exercise
1.3 Question 6
6. Write
the following as intervals :
(i)
{ x : x ∈ R, -4 < x ≤ 6 }
(ii)
{ x : x ∈ R, -12 < x < -10 }
(iii)
{ x : x ∈ R, 0 ≤ x < 7 }
(iv)
{ x : x ∈ R, 3 ≤ x ≤ 4 }
Solution:-
(i)
{ x : x ∈ R, -4 < x ≤ 6 } = ( -4,
]
(ii)
{ x : x ∈ R, -12 < x < -10 } =
(-12, -10 )
(iii)
{ x : x ∈ R, 0 ≤ x < 7 } = [ 0, 7 )
(iv)
{ x : x ∈ R, 3 ≤ x ≤ 4 } = [ 3, 4 ]
Class 11 Maths Chapter 1 Exercise
1.3 Question 7
7. Write
the following intervals in set-builder form :
(i)
( -3, 0 )
(ii)
[ 6, 12 ]
(iii)
(6, 12 ]
(iv)
[ -23, 5 )
Solution:-
(i)
( -3, 0 ) = {x : x ∈ R, -3 < x <
0 }
(ii)
[ 6, 12 ] = {x : x ∈ R, 6 ≤ x ≤ 12 }
(iii)
[ 6, 12 ] = {x : x ∈ R, 6 ≤ x ≤ 12 }
(iv)
[ -23, 5 ) = { x : x ∈ R, -23 ≤ x ≤ 5 }
Class 11 Maths Chapter 1 Exercise
1.3 Question 8
8. What
universal set (s) would you propose for each of the following :
(i)
The set of right triangles.
(ii)
The set of isosceles triangles.
Solution:-
(i)
For the set of right triangles, the
universal set can be the set of triangles or the set of polygons.
(ii)
For the set of isosceles triangles, the
universal set can be the set of triangles or the set of polygons or the set of
two-dimensional figures.
Class 11 Maths Chapter 1 Exercise
1.3 Question 9
9. Given
the sets A = { 1, 3, 5 }, B = { 2, 4, 6 } and C = { 0, 2, 4, 6, 8 }, which of
the following may be considered as universal set (s) for all the three sets A,
b and C
(i)
{ 0, 1, 2, 3, 4, 5, 6 }
(ii)
Φ
(iii)
{ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }
(iv)
{ 1, 2, 3, 4, 5, 6, 7, 8 }
Solution:-
(i)
It can be seen that A ⊂ { 0, 1, 2, 3, 4, 5,
6 }
B ⊂
{ 0, 1, 2, 3, 4, 5, 6 }
However,
C ⊄ { 0, 1, 2, 3, 4, 5, 6 }
Therefore,
the set { 0, 1, 2, 3, 4, 5, 6 } cannot be the universal set for the sets A, B
and C .
(ii)
A ⊄ Φ, B ⊄ Φ, C ⊄ Φ
Therefore, Φ cannot be the
universal set for the sets A, B and C.
(iii)
A ⊂ { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }
B ⊂ { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }
C ⊂ { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }
Therefore, the set { 0, 1,
2, 3, 4, 5, 6, 7, 8, 9, 10
} is the universal set for the sets A, B and C.
(iv)
A ⊂ { 0, 1, 2, 3, 4, 5, 6, 7, 8 }
B ⊂ { 0, 1, 2, 3, 4, 5, 6, 7, 8 }
However, C ⊄ {1, 2, 3, 4, 5, 6, 7, 8
}
Therefore, the set { 1, 2,
3, 4, 5, 6, 7, 8 } cannot be the universal set for the sets A, B, and C.
